Swiss Micros DM41X and Casio fx-CG 50: Minor Head Loss
Introduction
When a fluid, such as water, is flowing in a pipe system, energy is lost from the flow due to friction. This loss is known as head loss. The equation presented here is an equation to determine head loss occurring in pipe bends and joints (entrances and exits), which is categorized as minor loss. Minor losses are typically summarized with major losses to determine total head loss.
A formula for minor head loss is stated as:
h = C * v^2 / (2 * g)
where:
h = head loss (m)
C = coefficient
v = velocity of the fluid (m/s)
g = Earth’s gravity = 9.80665 m/s^2 (2 * g = 19.6133 m/s^2)
C = head loss coefficient (see table below)
Type # for C |
Value |
1. Sharp Exit |
1 |
2. Protruding Entrance |
0.8 |
3. Sharp Entrance |
0.5 |
4. Round Entrance |
0.1 |
If we fit the above table like so:
Type # for C |
Value |
1 |
1 |
2 |
0.8 |
3 |
0.5 |
4 |
0.1 |
Fortunately, the data above (and only the data above) can fit into the quadratic equation:
y = -0.05 * x^2 – 0.05 * x + 1.1 = -0.05 * (x^2 + x) + 1.1
where x is the type number and y is the corresponding coefficient. The assignment of type numbers is arbitrarily.
In the code for the DM41X, I use the polynomial to grab the required coefficient of the user’s choice, mapping choice 1 to C =1 for sharp exit, mapping choice 2 to C = 0.8 for protruding entrance, and so on. I got really lucky because the quadratic equation presented a perfect fit (r^2 = 1). In programming it can serve as alternative way to retrieve coefficient values (but the fit has to be perfect or near perfect with minor adjustments).
Swiss Micros DM41X Code: HEADLOS
(HP 41C compatible, no modules needed)
01 LBL^T HEADLOS
02 LBL 00
03 ^T SHARP EXIT
04 AVIEW
05 PSE
06 PSE
07 ^T PROTRUDING
08 AVIEW
09 PSE
10 PSE
11 ^T 3 SHARP ENT.
12 AVIEW
13 PSE
14 PSE
15 ^T 4 ROUND ENT.
16 AVIEW
17 PSE
18 PSE
19 ^T TYPE?
20 PROMPT
21 INT
22 STO 00 (comparison: reject if the entry is negative or greater than 5)
23 X<=0?
24 GTO 00
25 5
26 X<=Y?
27 GTO 00
28 RCL 00
29 X↑2
30 RCL 00
31 +
32 -20
33 /
34 1.1
35 +
36 ^T VEL. <M/S>?
37 PROMPT
38 X↑2
39 *
40 19.6133
41 /
42 ^T HEAD LOSS=
43 ARCL X
44 RTN
45 END
Casio fx-CG 50 Program HEADLOSS
Menu “TYPE”, “SHARP EXIT”, 1, “PROTRUDING”, 2, “SHARP ENTRANCE”, 3, “ROUND ENTRANCE”, 4
Lbl 1: 1 → C: Goto 5
Lbl 2: 0.8 → C: Goto 5
Lbl 3: 0.5 → C: Goto 5
Lbl 4: 0.1 → C: Goto 5
Lbl 5
“VELOCITY (M _| S)”? → V ( _| is the fraction character [ []/[] ] )
C × V² ÷ 19.6133 → H
“HEAD LOSS:”
H
Example
For the velocity, v = 10.5 ft/s ≈ 3.2004 m/s:
Type |
Head Loss (h) |
1. Sharp Exit |
0.5222 |
2. Protruding Entrance |
0.4178 |
3. Sharp Entrance |
0.2611 |
4. Round Entrance |
0.0522 |
Sources
Ajmera, Benna. “Engineering Formulas” Quick Study Academic. BarCharts, Inc. 2014
“Minor Losses in Pipes and Ducts“ Ansys. 2020. https://courses.ansys.com/wp-content/uploads/2020/09/Lesson-4-Minor-Losses-in-Pipes-and-Ducts-Handout.pdf Retrieved June 6, 2024.
Until next time,
Eddie
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